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Simplifying -16t2 + 240t + 16 = 0 Reorder the terms: 16 + 240t + -16t2 = 0 Solving 16 + 240t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(1 + 15t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(1 + 15t + -1t2)' equal to zero and attempt to solve: Simplifying 1 + 15t + -1t2 = 0 Solving 1 + 15t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + -15t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -15t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -15t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -15t + t2 = 0 + 1 -15t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -15t + t2 = 1 The t term is -15t. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15t + 56.25 + t2 = 1 + 56.25 Reorder the terms: 56.25 + -15t + t2 = 1 + 56.25 Combine like terms: 1 + 56.25 = 57.25 56.25 + -15t + t2 = 57.25 Factor a perfect square on the left side: (t + -7.5)(t + -7.5) = 57.25 Calculate the square root of the right side: 7.566372975 Break this problem into two subproblems by setting (t + -7.5) equal to 7.566372975 and -7.566372975.Subproblem 1
t + -7.5 = 7.566372975 Simplifying t + -7.5 = 7.566372975 Reorder the terms: -7.5 + t = 7.566372975 Solving -7.5 + t = 7.566372975 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = 7.566372975 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 7.566372975 + 7.5 t = 7.566372975 + 7.5 Combine like terms: 7.566372975 + 7.5 = 15.066372975 t = 15.066372975 Simplifying t = 15.066372975Subproblem 2
t + -7.5 = -7.566372975 Simplifying t + -7.5 = -7.566372975 Reorder the terms: -7.5 + t = -7.566372975 Solving -7.5 + t = -7.566372975 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = -7.566372975 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -7.566372975 + 7.5 t = -7.566372975 + 7.5 Combine like terms: -7.566372975 + 7.5 = -0.066372975 t = -0.066372975 Simplifying t = -0.066372975Solution
The solution to the problem is based on the solutions from the subproblems. t = {15.066372975, -0.066372975}Solution
t = {15.066372975, -0.066372975}
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